1. Introduction
As a core component, the health of rolling bearings has a huge influence on performance and safety of rotating machinery. Due to the complexity of the rolling bearings’ own structure and the harsh operating environment, they will inevitably have various faults, affecting the safety of rotating machinery. Therefore, the most effective way to avoid bearing faults causing significant economic losses and casualties is to carry out bearing fault diagnosis as soon as possible, and then repair and replace the damaged bearings [
1,
2,
3,
4,
5]. However, most methods for rolling bearing fault diagnosis are applied to constant speed conditions. In contrast, there has been less research on variable speed conditions, which are common conditions in rotating machinery [
6]. When rotating machinery is operating at variable speeds, the signal frequency will change greatly with time, the spectrum will become blurred and the signal will be strongly non-stationary, which makes the processing of vibration signals very challenging. Therefore, variable speed conditions are receiving extensive attention in the fault diagnosis research of rotating machinery.
Because the vibration signals are easily influenced by various noises during the acquisition process, it is an important premise to find an excellent signal preprocessing method to ensure the effective fault feature extraction. In this respect, many researchers have executed extensive research. Among them, Dragomiretskiy et al. [
7] proposed variational mode decomposition (VMD). However, the number of modes for VMD must be set in advance, which can make VMD lack of adaptability. In view of this shortcoming, researchers have studied adaptive variational mode decomposition (AVMD) and successive variational mode decomposition (SVMD), respectively [
8,
9]. Although these methods have reached some achievements, they must cooperate with other methods to complete signal preprocessing in the face of signal processing under variable speed conditions. However, the local mean decomposition (LMD) method proposed by Li [
10] can be directly applied to the nonlinear signals and then combined with the envelope spectrum to achieve the identification of bearing fault features. In order to improve the ability of the local mean decomposition (LMD), the ensemble local mean decomposition (ELMD), the complete ensemble local mean decomposition (CELMD) and the complete ensemble local mean decomposition (CELMDAN) have been proposed [
11,
12,
13]. Although these methods have achieved certain results, they still suffer from poor adaptability, modal mixing, end effects and so on. The robust local mean decomposition (RLMD) has solved the problems of the above methods [
14]. Moreover, RLMD combined with excellent time-frequency analysis tools can accurately extract features of variable speed signals [
15]. Therefore, this paper chooses RLMD as the tool to reduce noise.
After using RLMD, the useful fault information is extracted from the denoised signal by finding suitable features. Due to the inevitable friction during the bearing operation, the signal will show non-stationary and nonlinear characteristics. At this time, the entropy index is selected to extract different faults’ features. For example, Zhang et al. [
16] studied the multi-scale entropy (MSE). Li et al. [
17] proposed refined composite multiscale fuzzy entropy (RCMFE) to extract fault features, which are hidden in denoised signals. Omidvarnia et al. [
18] presented the concept of range entropy. Multiscale range entropy (MRE) was proposed on the basis of range entropy and successfully applied to bearing fault diagnosis [
19,
20]. Combined with the advantages of hierarchical analysis, the hierarchical range entropy (HRE) index was proposed [
21]. However, this index is only applied to constant speed conditions. Due to the complex time-varying modulation and spectral structure, it is very difficult for rolling bearing under variable speed conditions to extract fault features. To address the problem, a new index, called fractional hierarchical range entropy (FrHRE), is presented. Compared with the original HRE, FrHRE reflects the features of time, frequency and time-frequency domain at different scales. It can fully express the signal information for the sake of extracting multi-angle and deep-level signal features with variable speeds.
In addition to excellent fault feature extraction methods, it also needs to be matched with a good fault classifier so as to achieve high-precision classification under variable speed conditions. In recent years, many machine-learning methods have been applied on fault classification problems, for example support vector machine (SVM), extreme learning machine (ELM), kernel extreme learning machine (KELM), least squares support vector machine (LSSVM) and random forest (RF) [
22,
23,
24,
25,
26,
27,
28]. Due to the advantages of high classification accuracy and high calculation efficiency, RF is often applied to fault diagnosis for fault identification. Han et al. [
29] used the RF classifier to achieve accurate classification of rolling bearing faults. Vakharia et al. [
30] selected the RF to identify the ball bearing fault and realized high-precision classification. It can be observed that the classification performance of RF is excellent. However, there are many parameters that need to be tuned in RF model, which will affect the classification accuracy of RF to a certain extent. To solve this problem, researchers often use swarm intelligence optimization algorithms, which are very straightforward and easy to comprehend [
31]. For example, particle swarm optimization algorithm, artificial fish swarm algorithm and grey wolf optimization have been adopted to choose RF’s optimal parameters [
32,
33,
34]. The classification accuracy of the optimized RF is significantly improved. These swarm intelligence algorithms have been applied to RF’s optimization and attained certain results. However, their exploration and development capabilities are not very good. As a result, the hunter–prey optimization algorithm (HPO) has been proposed [
35]. The algorithm simulates the behavior of hunters and prey and has sufficient exploration and exploitation capabilities to find the optimal parameters adaptively. Therefore, this paper has applied HPO to the RF and adopted HPO-RF model for fault identification.
In conclusion, the paper presents a method for fault diagnosis with variable speed based on RLMD, FrHRE and HPO-RF. Notably, the paper make a summary on main contributions as follows:
A signal preprocessing method (RLMD) that can effectively remove noise from the variable speed signal is adopted.
A new feature extraction method applied to variable speed bearing signals, namely fractional hierarchical range entropy (FrHRE), is proposed in this paper.
This paper investigates an adaptive optimization-seeking fault identification model on the foundation of RF model. RF model parameters are globally optimized by iterative algorithm.
The rest of the paper is represented as detailed below:
Section 2 presents the basic theory of RLMD, FrFT, HRE, RF and HPO. The paper provides the explicit steps of FrHRE, presents the steps of HPO-RF and shows the steps of proposed method in
Section 3.
Section 4 analyzes the performance of RLMD, FrHRE and HPO-RF, respectively.
Section 5 discusses the experimental results. Finally, some conclusions are given in
Section 6.
2. Basic Theory
2.1. Robust Local Mean Decomposition (RLMD)
Robust Local Mean Decomposition (RLMD) can extract pure FM signal, envelope signal and their product function (PFs) from any complex signal to be analyzed, so that PFs can fully contain the multi-scale information of the original signal. Next, we will briefly introduce the process of extracting PFs by RLMD algorithm.
Step 1: Find the local extremes of , and calculate the smooth local mean and smooth local amplitude .
Step 2: The estimated zero-mean signal
and the FM signal
are calculated by Equation (1). The subscripts represent the
th PF and the
th sifting process.
Continue to repeat the above steps times with as a new signal until the sifting conditions proposed by RLMD () are met.
If the above sifting conditions are met, it can be proved that
is a pure FM signal, namely:
Step 3: Extract the remaining
from
.
can be expressed by Equation (3). Where
is the residual signal after
repetitions.
2.2. Fractional Fourier Transform (FrFT)
Performing FrFT on the signal is to rotate the signal counterclockwise on the time axis by
angle to the
axis and then perform the Fourier transform.
Figure 1 shows the process of using FrFT to rotate the
plane to the
plane.
The FRFT of signal
is defined as:
where
is the order of FRFT, which can change from 0 to 1, meaning that the signal changes from time domain to frequency domain gradually. Moreover, the kernel function of FRFT is:
Its transformation formula is shown in Equation (6), where the range of
is generally
.
2.3. Hierarchical Range Entropy (HRE)
HRE combines hierarchical decomposition and range entropy index, which can mine signal feature information from multi-scale aspects. Assuming a time series is given, the process of calculating its HRE is as follows:
First, we define two operators: the average operator
and the difference operator
, the formula is shown in Equation (7).
where
represents the low-frequency component of time series and
represents its high-frequency component [
36].
Secondly, given a integer
that is non-negative, there is only one vector
, which can represent the non-negative integer
, as shown in Equation (8).
Then, the operator should be used repeatedly in the hierarchical decomposition of time series. The hierarchical decomposition components can be attained by using Equation (9).
Finally, the range entropy is calculated for the hierarchical components obtained in the above procedures, and the hierarchical range entropy can be obtained.
2.4. Random Forest (RF)
RF is a machine learning algorithm, which can be used to classify, cluster and regress. The classification of RF is achieved by training base decision trees, generating models and using the comprehensive results of many decision trees to vote. The content of RF is roughly as follows [
37,
38]:
Step 1: Select
samples randomly as training sample set
by the bootstrap sampling method. A decision tree can be obtained by training using this sample set, as shown in Equation (11).
where
is the root node of the decision tree
, and
is the splitting criterion of decision tree.
Step 2: When each sample has features, select features randomly, and input the best one at each split node of the decision tree for splitting.
Step 3: Repeat the second step and continue splitting until all training samples of this node belong to the same class. There is no pruning behavior in the whole formation of the decision trees.
Step 4: Build multiple decision trees in accordance with the first three steps, which then can constitute a random forest.
Step 5: After the test samples are input into RF model, the class with the most classification results is selected as the final result. The classification decision model
is exhibited in Equation (12), where it represents the output tag variable and
is the indicative function.
2.5. Hunter–Prey Optimizer (HPO)
HPO is a new algorithm for optimizing proposed in 2021. These sections are described in detail below. First, Equation (13) are used to set the initial population
. Then, the fitness of each scheme is calculated by
.
where
represents the position of the hunter or prey,
represents the lower limits of the problem variables and
represents its upper limits.
is the number of these variables.
- (1)
The hunter search mechanism
In HPO, the hunter search mechanism is represented by Equation (14).
where
is the current position and
is the next position of the hunter.
is the mean value of all positions, that is,
.
is the balance parameter, which is calculated by
.
is the adaptive parameter, as Equation (15) shows.
is the position of the prey, which is shown in Equation (16).
- (2)
The prey escaped to a safe position
When the prey is attacked, it will escape to the safe position immediately. At this time, Equation (17) represents the update of the prey’s position.
where
and
are the current position and the next position of the prey, respectively.
stands for the optimal global location.
is a number randomly selected from
.
To solve the problem of how to distinguish the object represented by , HPO proposes an adjustment parameter (its value is 0.1) and a random number (its range is ). When , is regarded as the hunter. Otherwise, is regarded as the prey.
3. Proposed Method
3.1. Fractional Hierarchical Range Entropy (FrHRE)
A new feature extraction method, fractional hierarchical range entropy (FrHRE), is proposed by combining FrFT, hierarchical decomposition and range entropy in this paper. From different time-frequency perspectives, FrHRE retains the strength of multi-scale decomposition and adds high frequency components in different scales to make the extracted features contain more information.
Figure 2 is the flow chart for calculating FrHRE. Firstly, FrFT is applied to the time series to obtain the time-frequency components of different orders. The parameters are then initialized before calculating the HRE. Next, calculate the HRE for each time-frequency component. Finally, the output is HRE of each time-frequency component. That is, FrHRE is obtained.
3.2. Hunter–Prey Optimizer-Random Forest (HPO-RF)
In most cases, the number of base decision trees (ntrees) and minimum node-divided samples (n_splits) in RF are selected empirically, which may increase the classification error of RF. Therefore, this paper selects these two parameters as tuning objects to decrease the classification error as much as possible and enhance the RF’s accuracy. The optimization process of HPO-RF is as follows.
Input: training dataset, test dataset, the upper and lower limits of RF parameters and , the maximum number of iterations , the number of populations , the number of RF parameters and the adjustment parameter .
Output: the optimal solutions Target and TargetScore, which correspond to the optimal parameters and .
- (1)
According to the input parameters, the initial population is randomly set.
- (2)
Read the input training data and test data.
- (3)
Calculate the objective function of all members in the population and store the optimal fitness value found so far. (In this paper, we choose the sum of the classification error rates of training and test set as the fitness value, so the smaller fitness value is considered as the better one.)
- (4)
Update parameter: , , and set , , , , .
- (5)
Update the position of the current search agent using Equation (18).
3.3. The Proposed Method
Figure 3 exhibits the flow chart of the proposed method.
Step 1: The bearing signal is decomposed by RLMD. Moreover, the optimal components are chosen for signal reconstruction by using the principle of the maximum cross-correlation, which can reduce noise and highlight features to a certain extent.
Step 2: FrHRE is calculated separately for the denoised signal of each state, and a feature sample set is constructed. The parameters of FrHRE are set as: , the number of layers is 3, the embedding dimension is 2 and is 0.15 times of the sample’s standard deviation. Eleven components in the fractional Fourier domain (FrFD) can be obtained by performing FrFT of the corresponding order on the denoising signal. Where is the time domain of signal and is frequency domain, and the rest orders correspond to the signal’s time domain representation at different angles. Moreover, the range entropies at different levels are calculated for the obtained components in FrFD. Then the features in time, frequency and time-frequency domain are obtained.
Step 3: The samples of training and test are obtained by dividing extracted feature samples randomly, and the corresponding labels are given.
Step 4: The RF parameters that need to be optimized and their ranges are determined. Moreover, the input of HPO-RF are the sample set.
Step 5: RF is optimized using HPO. First, the parameters of the HPO are initialized. Secondly, the fitness values of all members in the initial population are computed (the classification error rate of the training and test samples is chosen as the fitness value). Next, the parameter values with the smallest fitness value are chosen as the current optimal parameters. After that, the parameters should continue to be update and agent’s current location should be searched for. Finally, when the maximum of iterations number is reached, the outputs are current parameters and the corresponding fitness values. If not, the target parameters should be updated and training continued until the iteration numbers reach the maximum.
Step 6: These obtained optimal parameters are input into RF for training and the classification of the test samples is completed.
6. Conclusions
The paper presents a novel model for variable speed bearing signal based on a combination of RLMD, FrHRE and HPO-RF.
In terms of signal preprocessing, the robust local mean decomposition (RLMD) is chosen for noise reduction of variable speed bearing signals. RLMD and other signal decomposition methods are applied to the simulated signal, respectively. By comparing their respective denoising effects, it is proved that RLMD is superior to other methods in eliminating noise, and it also has better performance in noise reduction.
In view of the difficulty of feature extraction for variable speed bearing signals, this paper introduces the fractional Fourier transform to improve it on the basis of hierarchical range entropy, and proposes a new method, namely fractional hierarchical range entropy (FrHRE). The comprehensiveness and stability of FrHRE in extracting entropy features are verified by using simulated noise signal.
Aiming at the problem that the parameters of random forest (RF) cannot be obtained adaptively, this paper improves the RF by using the hunter–prey optimization algorithm (HPO) in order to establish a random forest model with adaptive parameters. Compared with RF model improved by other optimization methods, the stability and accuracy of RF are superior.
The experimental data provided by Ottawa University validates that the proposed method can diagnose the faults effectively under variable speed conditions with good effects.
For the fault diagnosis that rolling bearing is in the working environment of variable speed, we can continue to improve FrHRE to complete higher precision feature extraction in the future. Furthermore, the proposed method should be used on other experimental data to further verify its universality.